Whereas a “microscopic" approach to monotonicity formulae leads to celebrated blow-up techniques
due to de-Giorgi ([1]}) and Fleming ([2]}),
and regularity theory in geometric measure theory (see [3]}, [4]}, [5]}, [6]}, [7]}, [8]}). Examining a macroscopic viewpoint of the above monotonicity formula, we derive vanishing theorems for vector bundle valued \(k\) -forms (see Theorem REF ). This macroscopic viewpoint also leads to the study of global rigidity phenomena in locally conformal flat manifolds (see Dong-Lin-Wei [9]}).
